Teaching mathematics as the contextual application of modes of mathematical enquiry
Anne Watson, University of Oxford
Bill Barton, The University of Auckland
Abstract
As teachers and educators we spent many hours in mathematics classrooms and had the privilege of being able to observe as well as participate. Our experiences, and our personal knowledge of mathematics, have led us to believe that mathematical modes of enquiry are a central part of teaching and learning mathematics. But a classroom is not a mini-mathematics laboratory in which students are apprentices. Rather, those teachers whose lessons make a significant difference to students’ understanding of mathematical ideas appear to adapt mathematical modes to the restricted frames of school mathematics.
We explored one of these frames, the preparation of teaching resources, to investigate our hypothesis about the central role of mathematical modes of enquiry. We set up an artificial resource preparation exercise amongst a group of knowledgeable mathematics educators and recorded their collaboration. We found that our personal mathematical modes were both transformed and transforming, and the results of this process were embedded into the resources we designed.
We argue that teachers’ fluency with mathematical modes is important in effective teaching. For teaching to mean anything, teachers must act in such a way that students learn something that they would not or could not have learned if the teacher had not been present. That is, the need to provide something that a textbook or annotated website cannot provide. Teachers’ fluency with mathematical modes of enquiry is the basis of their unique contribution.
One of Bill’s experiences
The syllabus I was using required five lessons on 2x2 matrices for my class of 14 year olds. We had looked at arrays and been through the operations +, –, x, ÷ with other matrices and 1x2 vectors. The final section is on matrices as transformations of the unit square: reflections, stretches, sheers, rotations. We do not quite finish so I use a little of the next lesson in a tight syllabus. In response to an invitation to the students to give me a random matrix so we can look at its effect, I get a 3x3 matrix suggested. Smart kid. The class appear to have understood the 2-dimensional concept, so I extend, draw a unit cube and watch as they quickly pick up the idea and stretch and reflect it in a plane. No problem—until the same child, flush with her success asks about a 4x4 matrix, with a smile, knowing there are only three dimensions. I seize the moment to demonstrate the power of mathematics to go beyond our experience and soon hypercubes are being reflected through 3-D space using the patterns of 2x2 and 3x3 reflections. The keen students take home homework on problems in 5 or 6 dimensions. But that lesson has been used up, and half the next one, and I am dreadfully behind my schedule. Why am I not feeling concerned and why do I remember that lesson as one of my best to this day?
One of Anne’s experiences
One student, a good mathematics graduate training to be a teacher, told me that he had expected to shut down his intellectual engagement with mathematics in order to teach at a lower level than normal. Instead he had found thinking about mathematics as a teacher every bit as mathematical and challenging as his first degree. An example occurred when thinking about preparing a lesson on straight line graphs, when he suddenly became aware that the schoolbook use of the term linear for functions of the form y=mx+c did not equate with his university use of the term linear to refer to functions for which λf(x) = f(λx). That is, keeping within the school syllabus, f(x)=3x–4 is not linear in the transformational sense. Heusedthis realisationto springboard a brief discussion with his school students of other mathematical meanings that appear to vary as you learn more mathematics, such as: multiplication not always making things bigger; translation being a different kind of symmetry from reflection, rotation and enlargement because it is not a matrix transformation; division not always being represented by sharing, and so on. Another teacher said that she relied on situated definitions, so that her students have to relate a definition of proportion in one context, such as ‘proportion of a whole’, to, say, proportionality as equality of two ratios.
The roles of mathematical modes of enquiry in teaching
Stories like these have led us to consider mathematical modes of enquiry: a teacher sparking off a student comment to make a wider point about mathematics and extend the students’ thinking when the moment was ripe, at the later cost of syllabus squeeze; a new teacher challenging himself with elementary material by thinking about mathematical definitions, assumptions, and implications; a lecturer becoming carried away with making connections and using new perspectives to re-view familiar material; new teachers treating a curriculum topic as an arena for comparing examples, definitions, assumptions, and implications. Again and again we observe, in ourselves and in others, that some of the best teaching and learning moments occur when mathematical modes of enquiry are invoked. We have come to believe that they are central to what a teacher does, and what a student is led to do.
In this chapter we test this belief by examining more closely the teacher act of preparing teaching resources. As will be seen, we gained confirming evidence of our belief. We learn that we need to understand betterhow teachers come to acquire these modes, and how and when they are invoked. What makes a lecturer go off on a mathematical tangent, and how can we tell whether that is useful for students? Why will a teacher insist on a detail of mathematical argument at one moment but allow an incomplete definition or use of a technical term in another? What is it about theirknowledge and awareness that links a particular mathematical mode to that teaching moment? In this chapter we cannot answer all these questions, but can begin a grounded investigation of teacher-thinking from a mathematical perspective.
To situate our investigation we comment briefly on current thinking about teachers’ mathematical knowledge andmore substantially on literature about mathematical enquiry.
Teacher knowledge
In the late 1980’s Shulman (1986, 1987) introduced the idea of pedagogical content knowledge in contrast to subject matter knowledge. His work responded to a local trend away from the generic professional development offered in the 1980s towards programmes that recognize the importance of enabling teachers to learn how to teach particular content. As Kennedy (1999) describes, teachers need to understand the ways students hold mathematical conceptions, to know what representations and analogies will be useful in teaching, and to understand developmental stages.
Since then, mathematical knowledge for teaching has often been theorised using the idea of acquisition of types of content knowledge for teaching. While such models might be useful for adding nuance to a continuum of pedagogical content knowledge and subject matter knowledge, and may thereby inform pre-service teacher development programmes, in our view they risk missing out the most crucial aspect of what a mathematics teacher does in relation to mathematics; teachers enact mathematics. In discussing mathematical knowledge for teaching, we can easily be drawn into a curriculum of items that a teacher needs to have learned: quadratic equations, differentiation, the history of negative numbers, stages in development of number awareness, common misconceptions, and so on. What is often missed is mathematical thinking and awareness. It is not just a question of what teachers know, but how they know it, how they are aware of it, how they use it. Perhaps this can be summed up as: what mathematical habits do they have? To be effective teachers, what do they need to do mathematically? It is the assumption of this paper that these mathematical modes of enquiry need to be deeply present in teaching. Further, we are unconvinced that we can attribute what they do to the personal possession of certain forms of knowledge.
Mathematical Modes of Enquiry
In March, 2008, the ICMI Centennial conference had a Working Group on the topic Disciplinary Mathematics and School Mathematics, at which questions were asked about the relationship between research mathematics and what happens in secondary classrooms. Initially it appeared that strongly differing orientations were being expressed: one the one hand it was argued that school mathematics had to be a “shadow” of the discipline, on the other that it was fundamentally different in context (Watson, 2008). However, a consensus did emerge that “students learn through reasoning that resembles mathematical thought” (Barton & Gordeau, 2008). It was noted that a significant difference between disciplinary and school mathematical experiences was the mediation of the teacher. This begs the question of how the teacher can best undertake the mediation; “working as a mathematician” was one answer to this question.
For us, any discussion of the mathematics involved in teaching has to start from understanding of what doing mathematics entails, and then seeing how this acts out in teaching. Otherwise there is a temptation to see teaching mathematics as to do with exercising knowledge, rather than as an arena for acting mathematically.
What is acting mathematically? Krutetskii’s (1976) seminal study of gifted Soviet mathematics students identified several common features. These students all had a tendency to:
•grasp formal structure;
•think logically in spatial, numerical and symbolic relationships;
•generalize rapidly and broadly;
•curtail mental processes;
•be flexible with mental processes;
•appreciate clarity and rationality;
•switch from direct to reverse trains of thought;
•memorize mathematical objects (1976).
These tendencies have been elaborated by Cuoco, Goldenberg and Mark (1996) to attach their specific manifestations in various branches of curriculum mathematics, and also extended to include the qualities of sustained niggling that bother mathematicians. Their characterisation of ‘habits of mind’ includes: pattern-sniffing, experimenting, visualising, forming conjectures, reasoning proportionally, loving systems, embracing unifying theories, looking at variance and invariance, extending meanings, thinking generally from examples and vice versa.
Sustained niggling is also described by Hadamard (1945) and extended to include moments after one has been totally engaged with a problem for a period of time, then relaxes to do something else, when insight occurs unexpectedly. This common experience reminds us that the natural ways in which the mind works includes reflection, organising, and seeking ways to compare and generalise experience. Mason puts some structure on ‘sustained niggling’ (Mason, 1988) by focusing on stages and states of mathematical thinking. His inspiration came from Polya’s (1962) classic work on problem-solving, encapsulating Polya’s extensive list of the many strategies on which mathematicians can call. These have been sloganised as ‘specialise, generalise, conjecture, convince’, but their common use is as instructions rather than as descriptions of behaviour. This sometimes leads to an assumption that these actions happen in a given order. It is more common for mathematical thinking to roam between and within these approaches. It is also worth noting that ‘specialise’ implies a special choice of examples, rather than using examples as data for inductive purposes. We mention this here because purposeful generation and use of examples is also a major feature of being mathematical and of course one that characterises good planning and teaching. For example, in a lesson about probability the teacher offered questions in which P(r) + P(¬r) = 1 emerged as an conjecturethat was obvious to many learners, followed by a question in which P(r)=1 and a following question in which P(¬r) = 1.Creating and using examples to structure generality requires that teachers see what they are teaching in terms of generalities rather than techniques. Using extreme and special examples is a common mode of enquiry to see how far a conjecture holds up. This teacher clearly understood this and tried to communicate it to her students: ‘Look’, she said ‘the mathematics is telling you something’.
Of course one cannot be mathematical without the specific intellectual toolkit and repertoire of mathematics. The ‘habits of mind’ model includes some of these, such as using understandings of probability, representation and generalisation in the example above. Simon (2006) identified key developmental understandings of mathematics not as first order knowledge, but as foundations for learning other ideas. We would see these key understandings as threads that run throughout mathematics, so that the ways in which we read mathematical situations are profoundly and lastingly influenced by them. For example, understanding number multiplicatively as a first resort, not as something to be used if additive models fail, is a key understandingin much secondary and tertiary mathematics; understanding functions as mathematical objects, rather than as algebraic representations of data sets, is key to understanding much higher mathematics. Silverman and Thompson (2008) show that merely being offered situations in which these are made apparent as useful ways to view mathematics is not guaranteed to lead to good mathematics teaching. We want to develop the reverse story: how do teachers who have, over time, developed significant ways of thinking about and interpreting mathematics, bring that experience and knowledge to bear on pedagogic tasks?
An artificial teacher activity
In order to explore our belief that mathematical modes of enquiry are central to effective teacher activity, we set up an artificial teacher activity, that of developing teacher resources, and asked two other experienced mathematics educators to join us. All four of us regard ourselves as mathematicians in our habits. We hoped to examine what knowledgeable mathematics educators do as they think about presenting students with mathematical situations. We took two starting-points and gave ourselves the task of using them to devise a teaching situation. We recorded the discussion and then analysed it to identify the mathematical practices and repertoire implicit and explicit in our responses. The use of the behaviour of experts to explore what is possible has strong precedents (for example, Carlson & Bloom 2005). We do not claim that what we did is what novices would do, nor that ours was the best, or only, possible behaviour; rather we are using this shared task to see what can be said about the role of mathematical expertise in typical planning discussions.
The first stimulus was a page of exercises from a school textbook chosen because it exemplified ‘dry’ problems of pure mathematics that are very familiar to any secondary mathematics teacher. The second stimulus was that day’s newspaper. The task for the team of four was to describe the possibilities they could see in these stimuli for a teacher faced with creating a lesson based on them. We acknowledge that this is a false situation. It is very rare that a teacher would have the luxury of two hours with three interested colleagues to create two lessons. Lessons are usually created within a continuous curriculum, and with certain aims. For this exercise we assumed the content aims of the author of the textbook page, and also assumed that exploring the mathematics of an ‘everyday’ issue from the newspaper could be a lesson aim in itself.
Our approach was to first identify the range of potential mathematics afforded by these artefacts, according to our mathematical knowledge. In addition we agreed to discuss, after the event, what mathematical knowledge we used and how a teacher might recognise the potential that we had recognised and bring it to realisation in a classroom.
Stimulus 1Problems about inverse proportion (see Appendix 1)
This textbook page introduced inverse proportionality, expressing this relationship in a variety of ways, including two which offered interplay between data sets and algebraic representations. A range of letters as variables was used, and the independent variable was itself a function in some of the questions.
Our initial responses may be described as alerts: Pedro[1] noted that the symbol is not universal, and, for example, is not used in Portugal; Bill noted that the term “inverse” has multiple mathematical meanings that students of this level would know (for example, the inverse operation, and ‘invert and multiply’); Anne noted that particular letters used on the page, such as t, L and v, have common contextual meanings in mathematics and science that were not preserved in these examples; and John noted that the two ways of expressing inverse proportion algebraically (implied in Qn.1a) and (implied in Qn.2) are not obviously equivalent. All four of us felt that both these expressions of relationship are fundamental to a fluent understanding of, and recognition of, inverse proportionality—too important to be offered only as implications, as in Qn.1 and Qn.2.
The group then began to respond in more detail to features of the set of exercises. For example, Qn.4 essentially asked for the factors of 12, and could be completed without thinking about inverse proportion. Another feature noted was that all the numbers on the page are simple integers, or simple fractions, and have simple multiplicative relationships. The discussion quickly moved into a suggestion by Pedro that Qns. 7 & 8 would be interesting if the table did not include any matched pair. For example, instead of
y / 2 / 4 / 1/4z / 8 / 16
one could offer
y / 4 / 1/4z / 8 / 16
Such an exercise would not only make the question more open, but students working with it would be led naturally into the concept of the constant of proportionality, k. We wondered if the necessity to understand an expression such as t2to be a variable in itself, as is offered in Qn.8, detracted from the main idea of inverse proportion, or encouraged a focus on ‘inverseness’. The extensive discussion ended up with agreement that a whole series of lessons could be constructed so that students would develop for themselves the concept of inverse proportion by engaging with suitably constructed and sequenced tasks that either avoided completion by merely templating numbers, or that challenged such completion by affording cognitive conflict. However, we agreed that the page as a whole attempted to avoid the possibility that learners would get locked into simplistic assumptionsabout the relationship, and did offer the potential for a complex engagement with the concept approached from several different perspectives, using different symbolisations and also within composite functions.